Editing Andrew Wiles (section)

==Proof of Fermat's Last Theorem==
{{main|Wiles's proof of Fermat's Last Theorem}}
From 1988 to 1990, Wiles was a Royal Society Research Professor at the [[University of Oxford]], and then he returned to Princeton.
From 1994 to 2009, Wiles was a [[Eugene Higgins|Eugene Higgins Professor]] at Princeton.

Starting in mid-1986, based on successive progress of the previous few years of [[Gerhard Frey]], [[Jean-Pierre Serre]] and [[Ken Ribet]], it became clear that [[Fermat's Last Theorem]] (the statement that no three [[positive number|positive]] [[integer]]s {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} satisfy the equation {{math|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} for any integer value of {{math|''n''}} greater than {{math|2}}) could be proven as a [[corollary]] of a limited form of the [[modularity theorem]] (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture").<ref name="Darmon 1999"/> The modularity theorem involved elliptic curves, which was also Wiles's own specialist area, and stated that all such curves have a modular form associated with them.<ref>{{cite web|last=Brown|first=Peter|title=How Math's Most Famous Proof Nearly Broke|url=http://nautil.us/issue/24/error/how-maths-most-famous-proof-nearly-broke|publisher=Nautilus|access-date=16 March 2016|date=28 May 2015|archive-date=15 March 2016|archive-url=https://web.archive.org/web/20160315201345/http://nautil.us/issue/24/error/how-maths-most-famous-proof-nearly-broke|url-status=dead}}</ref><ref name="NYT-20220131">{{cite news |last=Broad |first=William J. |authorlink=William Broad |title=Profiles in Science – The Texas Oil Heir Who Took on Math's Impossible Dare – James M. Vaughn Jr., wielding a fortune, argues that he brought about the Fermat breakthrough after the best and brightest had failed for centuries to solve the puzzle. |url=https://www.nytimes.com/2022/01/31/science/james-vaughn-fermat-theorem.html |date=31 January 2022 |work=[[The New York Times]] |accessdate=2 February 2022 }}</ref> These curves can be thought of as mathematical objects resembling solutions for a torus' surface, and if Fermat's Last Theorem were false and solutions existed, "a peculiar curve would result". A proof of the theorem therefore would involve showing that such a curve would not exist.<ref name=nyt/>

The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove.<ref name="Singh">[[Simon Singh]] (1997). ''[[Fermat's Last Theorem (book)|Fermat’s Last Theorem]]''. {{ISBN|1-85702-521-0}}</ref>{{rp|203–205, 223, 226}} For example, Wiles's ex-supervisor [[John H. Coates|John Coates]] stated that it seemed "impossible to actually prove",<ref name="Singh"/>{{rp|226}} and [[Ken Ribet]] considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."<ref name="Singh"/>{{rp|223}}

Despite this, Wiles, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for [[Frey's elliptic curve|Frey's curve]].<ref name="Singh"/>{{rp|226}} He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.<ref name="Singh"/>{{rp|229–230}}

Wiles' research involved creating a [[proof by contradiction]] of Fermat's Last Theorem, which Ribet in his [[Ribet's theorem|1986 work]] had found to have an elliptic curve and thus an associated modular form if true. Starting by assuming that the theorem was incorrect, Wiles then contradicted the Taniyama–Shimura–Weil conjecture as formulated under that assumption, with Ribet's theorem (which stated that if {{math|''n''}} were a [[prime number]], no such elliptic curve could have a modular form, so no odd prime counterexample to Fermat's equation could exist). Wiles also proved that the conjecture applied to the special case known as the [[semistable elliptic curve]]s to which Fermat's equation was tied. In other words, Wiles had found that the Taniyama–Shimura–Weil conjecture was true in the case of Fermat's equation, and Ribet's finding (that the conjecture holding for semistable elliptic curves could mean Fermat's Last Theorem is true) prevailed, thus proving Fermat's Last Theorem.<ref>{{Citation|last=Stevens|first=Glenn H.|author-link=Glenn H. Stevens|url= https://math.bu.edu/people/ghs/papers/FermatOverview.pdf|title=An Overview of the Proof of Fermat's Last Theorem|date=n.d.|publisher=Boston University}}</ref><ref>{{Citation|last=Boston|first=Nick|url= https://people.math.wisc.edu/~nboston/869.pdf|date=Spring 2003|title=Proof of Fermat's Last Theorem|publisher=University of Wisconsin–Madison}}</ref><ref name="Darmon 1999"/>

In June 1993, he presented his proof to the public for the first time at a conference in Cambridge. [[Gina Kolata]] of ''[[The New York Times]]'' summed up the presentation as follows:
{{Blockquote|He gave a lecture a day on Monday, Tuesday and Wednesday with the title "Modular Forms, Elliptic Curves and Galois Representations". There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.<ref name=nyt>{{cite news|last=Kolata|first=Gina|author-link=Gina Kolata|title=At Last, Shout of 'Eureka!' In Age-Old Math Mystery|url=https://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html|access-date=21 January 2013|newspaper=[[The New York Times]]|date=24 June 1993|archive-url= https://web.archive.org/web/20231120054908/https://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html|archive-date=20 November 2023}}</ref>}}
In August 1993, it was discovered that the proof contained a flaw in several areas, related to properties of the [[Selmer group]] and use of a tool called an [[Euler system]].<ref name="Faltings 1995" /><ref name="Cipra 1995">{{cite journal| last=Cipra|first=Barry Arthur |author-link=Barry Arthur Cipra|title=Princeton Mathematician Looks Back on Fermat Proof|journal=[[Science (journal)|Science]]|volume=268|date=1995|issue=5214 |pages=1133–1134 |doi=10.1126/science.268.5214.1133 |pmid=17840622 |bibcode=1995Sci...268.1133C }}</ref> Wiles tried and failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing—rather than closing—this area came to him on 19 September 1994, when he was on the verge of giving up. The circumvention used [[Galois representations]] to replace elliptic curves, reduced the problem to a [[class number formula]] and solved it, among other matters, all using [[Victor Kolyvagin]]'s ideas as a basis for fixing [[Matthias Flach (mathematician)|Matthias Flach]]'s approach with Iwasawa theory.<ref name="Cipra 1995" /><ref name="Faltings 1995">{{cite journal | last=Faltings|first=Gerd|author-link=Gerd Faltings|title=The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles| publisher=[[American Mathematical Society]]| date=July 1995|volume=42|issue=7 | url= https://www.ams.org/notices/199507/faltings.pdf|page=743-746 |journal=Notices of the AMS | access-date=1 August 2024}}</ref> Together with his former student [[Richard Taylor (mathematician)|Richard Taylor]], Wiles published a second paper which contained the circumvention and thus completed the proof. Both papers were published in May 1995 in a dedicated issue of the ''[[Annals of Mathematics]].''<ref name="Wiles 1995">{{Cite journal|last=Wiles|first=Andrew|date=May 1995|title=Issue 3|jstor=i310703|journal=[[Annals of Mathematics]]|volume=141|pages=1–551}}</ref><ref>{{cite web|title=Are mathematicians finally satisfied with Andrew Wiles's proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?|url=http://www.scientificamerican.com/article/are-mathematicians-finall/|work=[[Scientific American]]|access-date=16 March 2016|date=21 October 1999}}</ref>